A quantum accounting and detective quantum efficiency analysis for video-based portal imaging

Transcription

1 A quantum accounting and detective quantum efficiency analysis for video-based portal imaging Jean-Pierre Bissonnette a),b) London Regional Cancer Centre, 790 Commissioners Rd. E., London, Ontario, N6A 4L6 Canada, and Department of Medical Biophysics, University of Western Ontario, London, Ontario, N6A 5C1 Canada I. A. Cunningham Imaging Research Laboratories, John P. Robarts Research Institute, P.O. Box 5015, 100 Perth Drive, London, Ontario, N6A 5K8 Canada; Department of Diagnostic Radiology, Victoria Hospital, London Ontario, N6A 4G5 Canada; and Departments of Radiology and Nuclear Medicine and of Medical Biophysics, University of Western Ontario, London, Ontario, N6A 5C1 Canada David A. Jaffray Department of Radiation Oncology, William Beaumont Hospital, Royal Oak, Michigan A. Fenster Imaging Research Laboratories, John P. Robarts Research Institute, P.O. Box 5015, 100 Perth Drive, London, Ontario, N6A 5K8 Canada and Departments of Radiology and Nuclear Medicine and of Medical Biophysics, University of Western Ontario, London, Ontario, N6A 5C1 Canada P. Munro London Regional Cancer Centre, 790 Commissioners Rd. E., London, Ontario, N6A 4L6 Canada and Departments of Oncology, Medical Biophysics, and of Physics, University of Western Ontario, London, Ontario, N6A 5C1 Canada Received 23 June 1995; accepted for publication 17 March 1997 The quality of images generated with radiographic imaging systems can be degraded if an inadequate number of secondary quanta are used at any stage before production of the final image. A theoretical technique known as a quantum accounting diagram QAD analysis has been developed recently to predict the detective quantum efficiency DQE of an imaging system as a function of spatial frequency based on an analysis of the propagation of quanta. It is used to determine the quantum sink stage s stages which degrade the DQE of an imaging system due to quantum noise caused by a finite number of quanta, and to suggest design improvements to maximize image quality. We have used this QAD analysis to evaluate a video-based portal imaging system to determine where changes in design will have the most benefit. The system consists of a thick phosphor layer bonded to a1mmthick copper plate which is viewed by a T.V. camera. The imaging system has been modeled as ten cascaded stages, including: i conversion of x-ray quanta to light quanta; ii collection of light by a lens; iii detection of light quanta by a T.V. camera; iv the various blurring processes involved with each component of the imaging system; and, v addition of noise from the T.V. camera. The theoretical DQE obtained with the QAD analysis is in excellent agreement with the experimental DQE determined from previously published data. It is shown that the DQE is degraded at low spatial frequencies 0.25 cycles/mm by quantum sinks both in the number of detected x rays and the number of detected optical quanta. At higher spatial frequencies, the optical quantum sink becomes the limiting factor in image quality. The secondary quantum sinks can be prevented, up to a spatial frequency of 0.5 cycles/mm, by increasing the overall system gain by a factor of 9 or more, or by improving the modulation transfer function MTF of components in the optical chain American Association of Physicists in Medicine. S Key words: quantum accounting diagram QAD, detective quantum efficiency DQE, portal imaging I. INTRODUCTION Medical x-ray imaging systems are generally designed to produce images having the highest quality possible for a specified dose. One measure of system performance is the detective quantum efficiency DQE, which is a measure of how efficiently an imaging system transfers the signal-tonoise ratio SNR in the incident radiation beam. This SNR describes the ratio of the number of x-ray quanta to the random fluctuation in this number of quanta. The DQE is reduced if the imaging system does not detect all of the incoming x rays, adds noise to the image i.e., noise due to electronic components or to variations in the intensity of bursts of light produced by the x-ray detector 1, or suffers 815 Med. Phys. 24 (6), June /97/24(6)/815/12/$ Am. Assoc. Phys. Med. 815

2 816 Bissonnette et al.: Quantum accounting and detective quantum efficiency 816 from a low gain stage that results in an insufficient number of quanta at the final stage of the image-formation process to maintain the necessary SNR. The analysis described here does not account for fixed-pattern noise 2 or noise generated by variations in light output caused by localized energy deposition and optical transport e.g., noise described by Nishikawa et al. 1. A theoretical analysis of imaging systems, known as a quantum accounting analysis, has been introduced recently by Cunningham et al. 3 to help design imaging systems. In this theory, which is based on expressions for the propagation of noise and signal derived by Rabbani et al., 4 6 the average quantum fluence, modified by the modulation transfer function MTF squared, is determined for each stage of an imaging system and displayed on a quantum accounting diagram QAD, 3 similar to a nomogram. 7 Designers can use a QAD to identify the stage with the fewest quanta at any specified frequency i.e., a spatial-frequency-dependent quantum sink which generally limits the SNR of the imaging system at that frequency. Therefore, designers can use the approximate QAD approach to optimize the performance of an imaging system by ensuring that the quantum sink occurs only at the x-ray detection stage for any specified spatial frequency. A conventional zero-frequency version of this approach has been in use for at least 45 years, 8 but ignoring the frequency dependence has been shown to give misleading results. 3,9 One area of medical imaging where the QAD theory can be useful is portal imaging. In a previous study, a zerofrequency analysis of a video-based portal imaging system was performed; 10 however, this type of simplified analysis cannot account for the effects of improving spatial resolution or reducing additive noise. In this paper, we analyze a T.V. camera-based portal imaging system 10 using QAD formalism to determine the noiselimiting processes of this imaging system and to determine which system parameters could be modified to improve the overall imaging performance. Furthermore, we present a theoretical calculation, derived from this QAD analysis, of the spatial-frequency-dependent DQE of this particular imaging system and compare the theoretical DQE with an experimental DQE determined using previously published data. 11 II. BACKGROUND A. QAD analysis Characterization of the portal imaging system using the quantum accounting analysis theory 3 6 requires representing the system as a serial cascade of multiple stages, where the number of quanta leaving each stage constitutes an effective input to the subsequent stage. This type of analysis can be used to describe linear and shift-invariant systems. The average number of quanta per unit area ( ) as well as the noise-power spectrum NPS assuming stationary noise can be determined at each stage of the image-formation process, and the number of quanta at any stage can be related to the average number of incident x-rays. In this theory, quanta are propagated through two types of stages: i Gain processes, where the number of quanta exiting a stage is related to the number of quanta entering by an average gain factor e.g., amplification, collection efficiency, detection efficiency, and associated gain variance ; and ii stochastic spatial spreading processes, where the number of quanta is conserved. Stochastic spatial spreading occurs when quanta are dispersed randomly into a spatial distribution with a probability given by the point-spread function, which is related to the MTF. 3,4 Stochastic spreading has also been described mathematically as a stochastic convolution process, and results from the statistical properties of the quanta e.g., light photons undergoing the spatial spreading. 12 The order of each imageforming stage is important, and, in the quantum accounting analysis, each stage can be either a gain or a stochastic spreading process, but not both. Therefore, any physical process which involves both gain and spread must be represented by two separate stages a gain stage and a spread stage in the quantum accounting analysis. 1. Gain stages When the ith stage consists of a gain only, the propagated mean fluence ( i) and gain Poisson excess ( gi ) are 3 i g i 1 i where i 1 2 gi and gi g 1, i is the mean number of quanta per unit area incident on stage i, g i is the average fluence gain associated with stage i, and 2 gi is the variance in this gain. The Poisson excess represents the relative amount by which the gain variance is in excess of a gain which follows Poisson statistics. The gain may represent an amplification or an interaction probability. The latter is a binary selection process described by a probability g that i a given input quantum is propagated to the next stage. As a result, the outcome can be 1 or 0 only. For binary selection processes, the Poisson excess is gi g, 3 i as a consequence of the binomial theorem. 2. Stochastic spreading stages A spreading process is characterized by the MTF of the stage, T i ( ). When the ith stage is a spreading process, the propagated mean fluence and gain Poisson excess are 3 i i 1 and gi 1, 2 since the spreading process involves no gain. Rather, a spreading process involves a random redistribution of individual quanta according to the point-spread function associated with the spreading process. 3,4,12 B. DQE analysis The DQE of the portal imaging system can be derived, in terms of the signal and noise transfer characteristics of the system, using two approaches. The first approach makes use of measured values for the NPS and the MTF of the imaging system as a whole to calculate the DQE. This approach will 1

3 817 Bissonnette et al.: Quantum accounting and detective quantum efficiency 817 be referred to throughout the paper as the experimental approach. In the second approach, the DQE is calculated using the QAD analysis, where the mean gains, MTF s, and Poisson excesses from individual stages are propagated. 1. Experimental approach In terms of the NPS and MTF of an imaging system, the DQE can be written as 3 DQE Ḡ2 T 2 out 0, 3 S out where is 0 the mean x-ray fluence of the input radiation beam in units of 1/mm 2 ), S out ( ) is the NPS of the image in units of 1/mm 2 ), Ḡ is the average gain of the entire imaging system, and T out ( ) is the MTF of the entire imaging system. Note that Eq. 3 is slightly different from the more usual form, where is 0 in the denominator. This is because our notation conforms to the notation of Cunningham et al QAD approach The spatial-frequency-dependent DQE can also be expressed in terms of the average gain and MTF of individual stages, as well as any additive noise introduced in a stage. From Eqs. 1 and 2, the expression for the DQE derived by Cunningham et al. 3 is DQE 1,M M 1 i g i T i 2 S ai / i P i, 4 where S ai ( )/ i is the relative additive noise in the ith stage and P i ( )is the product of the gains and squared MTF s up to and including the ith stage, given by P i j 1 i T g j j 2. 5 FIG. 1. Schematic of the T.V. camera-based portal imaging system. Equation 5 describes the product of the mean gain and squared MTF of every stage up to and including the ith stage and can be interpreted as an effective number of quanta xrays, light quanta or photoelectrons which propagate the image signal through the stages of the system, at each spatial frequency. A plot of P i ( ), as a function of stage number i, yields the quantum accounting diagram. The stage with the lowest P i ( ) value corresponds to the limiting spatialfrequency-dependent quantum sink, and is generally the noise-determining stage. In general, the DQE given by Eq. 4 is degraded at any stage where P i ( ) is close to or less than unity. Therefore, this approach can be used to determine which stage s are the noise-limiting stages, and what changes in system design are required to prevent secondary quantum sinks and thus maximize image quality. The DQE of an imaging system can therefore be described both by: i Its overall signal and noise transfer properties Eq. 3 ; and ii the signal and noise transfer properties of each stage in a linear system model Eqs. 4 and 5. In this article, the experimental DQE obtained using Eq. 3 and previously published data 11 is compared with the theoretical DQE obtained using Eq. 4. III. SYSTEM DESCRIPTION The portal imaging system developed at the London Regional Cancer Centre, shown schematically in Fig. 1, consists of a T.V. camera Video Optics, V1509 B camera which detects the light produced by the copper plate/ phosphor screen x-ray detector through a 45 mirror and a large aperture lens Fujinon, 50 mm, F/0.7 with a demagnification factor of ,11,13 The video signal is then digitized by an 8-bit frame-grabber Infimed, Liverpool, NY installed on a Sun Sparcstation 2 Sun Microsystems, Mountain View, CA. The copper plate/phosphor screen detector consists of a 360 mg/cm 2 thick layer of Gd 2 O 2 S:Tb phosphor directly bonded to a1mmthick copper plate Eastman Kodak, Rochester, NY. In order to minimize the contribution of the T.V. camera noise to the portal images, the camera is operated in a pulse-progressive mode, where the light emitted from the phosphor screen is accumulated on the lead-oxide target of the T.V. camera Philips Components, XQ2182 tube throughout a short irradiation e.g., from 0.25 to 2.0 s. The DQE and QAD analyses, as do all NPS-based methods, assume linear, shift-invariant systems with stationary noise processes. The MTF and NPS for the central region ( cm 2 ) of images obtained with the portal imaging system have been measured previously using a6mvx-ray beam. 11 In this region, the portal imaging system can be assumed to be linear 10 and shift-invariant. The latter condition is closely approximated in the central quadrant of images obtained with the portal imaging system, where both lens vignetting and off-axis lens aberrations affect image quality minimally. Also, the random noise processes present in T.V. camera-based systems can be assumed to be stationary and ergodic in T.V.-based radiographic systems. 14 Since all the measurements involving the lens and T.V. camera have been performed on the central axis i.e., MTF or on the central quadrant of images i.e., NPS, the portal imaging system can be modeled, using the QAD approach, for a 6 MV beam to enable a comparison of the theoretical and measured DQE.

4 818 Bissonnette et al.: Quantum accounting and detective quantum efficiency 818 FIG. 3. Absorbed energy distribution for the copper plate/phosphor screen detector, when irradiated by a 6 MV x-ray beam. This distribution is used to establish g 1, g 3, and their respective Poisson excesses. The average energy deposited in the phosphor layer per x ray which actually deposit energy in the phosphor layer i.e., 0.44 MeV results, on average, in optical quanta being produced. The abrupt change in the AED at 0.25 MeV is caused by the poor resolution of the 6 MV x-ray spectrum used to calculate the AED. FIG. 2. Block diagram showing the ten image-forming stages which have been used to analyze the T.V. camera-based system using the QAD theory. The QAD theory requires that the gain, Poisson excess, and MTF of each stage be determined. IV. QAD ANALYSIS For the purpose of the QAD analysis, the system has been divided into ten stages consisting of gain or spreading processes only, as shown in Fig. 2. This analysis requires that the gain, Poisson excess, and MTF of each stage described in this section be determined. We have performed this analysis in the direction perpendicular to the scan lines of the T.V. camera only i.e., vertical direction. While a full two-dimensional calculation is more comprehensive, our previous work has shown that the MTF of the portal imaging system is uncertain in the direction parallel to the scan lines of the T.V. camera i.e., horizontal direction since the design and adjustment of the camera electronics i.e., bandwidth of the preamplifier affect the shape of the line-spread function LSF in this direction. 11 The MTF measured in the perpendicular direction is generally worse than in the parallel direction, and hence our QAD and DQE analysis represents a worst-case estimate. A. QAD analysis 1. Stage 1: Detection of primary x-ray quanta, g 1 The first stage is the deposition of x-ray energy in the x-ray detector. Only those x-ray photons which interact with the copper plate/phosphor screen detector and deposit energy in the phosphor screen produce optical quanta which contribute to the portal image. The probability (g 1 ) that an incident x ray interacts in this way has been calculated using a computer program based on the EGS4 Monte Carlo system. This program, which has been described by Jaffray et al., 15 is used to generate an absorbed energy distribution AED in a specified material. The AED describes the number of incident x rays photon histories which deposit an energy between E and (E E) in the phosphor layer of the x-ray detector as a function of energy E. One such AED is shown in Fig. 3. As far as our analysis goes, the only x-ray photons which are considered in the AED are those which actually deposit energy i.e., E 0) in the phosphor layer. This is because x rays which interact with the detector but do not deposit energy in the phosphor layer do not generate the optical quanta which are necessary to form an image. The value of g 1 is determined as the area of the AED i.e., the total number of energy deposition events divided by the total number of x-ray photons incident on the detector. The AED was calculated ten times, using independent sequences of random numbers, and the values of g 1 obtained from each of the ten AED s were averaged to reduce the uncertainty in the value of g 1. Each AED was calculated using 10 6 photon histories and the 6 MV x-ray spectrum of Kubsad et al. 16 The useradjustable parameters which affect the transport of highenergy particles in the EGS4 program e.g., the corrected parameter reduced electron step transport algorithm, 17,18 the minimum total energy of photons and electrons which are transported ECUT, PCUT, and the energy thresholds for the creation of secondary photons and electrons AE, AP were set as described by Jaffray et al. 15 For the 6 MV spectrum impinging on our x-ray detector a 1 mm copper plate and a 360 mg/cm 2 gadolinium oxysulfide phosphor screen, we estimate that approximately 3.5% of the incident x-ray photons deposit energy in the phosphor screen i.e., g ). 2. Stage 2: Spread of high energy particles in the phosphor screen, T 2 ( ) Spread of signal in the copper plate/phosphor screen detector occurs because of migration of high energy particles

5 819 Bissonnette et al.: Quantum accounting and detective quantum efficiency 819 i.e., scattered x rays, electrons set in motion by x-ray interactions in the x-ray detector, and bremsstrahlung in both the metal plate and the phosphor screen as well as because of light spread in the phosphor screen. 19 To separate these two effects and determine the spread due to high-energy particles alone, the following technique was used. Two steel blocks 60 cm thick with a 72 mm groove cut along the length of one block were clamped together to form a narrow slit. This slit collimator was carefully aligned with the x-ray source to produce a very narrow beam. The beam was used to irradiate the copper plate/phosphor screen, which was covered with a thin 60 m sheet of opaque, black plastic to prevent optical photons from reaching Kodak EM-1, single-emulsion film which was placed in close contact with the plastic film. Thus, the LSF recorded by these films was due to the interaction of high-energy particles with the film emulsion. These films were processed in a Kodak M6B X-Omat film processor. The LSF s recorded on two series of films were digitized twice with a Perkin Elmer PDS scanning microdensitometer, first using an aperture which was 20 m and then with an aperture which was 10 m. A small sampling increment 20 m first, 5 m second was used to minimize the effect of aliasing in the measurement. The measurements yielded similar MTF curves, suggesting little aliasing was present. To determine the tails of the LSF accurately, five films were obtained, each exposed to a different x-ray dose ranging between 250 and monitor units, where one monitor unit 1 cgy dose to the isocentre for a cm 2 field at a depth of 5 cm in water. The film nonlinearities were corrected by using the characteristic curve measured with the scanning densitometer for EM-1 film exposed to high-energy particles. The resulting LSF was processed using standard techniques used for screen/film combinations. 11,13,19 21 The measured MTF of the copper plate/phosphor screen due to the transport of high energy particles, T 2 ( ), is shown in Fig Stage 3: Generation of optical quanta in phosphor, g 3 The AED s described in Sec. IV A were also used to determine the gain associated with the third stage (g 3 ),which is the conversion of x-ray quanta to optical quanta. The average number of optical quanta generated per x-ray which actually deposit energy in the phosphor layer is 22 g 3 E ab E, 6 opt where is E ab the mean energy absorbed in the phosphor layer per photon history, is the energy conversion efficiency i.e., the fraction of deposited energy converted to optical energy, and E is opt the mean energy of the optical quanta produced. For Gd 2 O 2 S:Tb, is 0.15 and is E opt 2.3 ev. 23 The light emitted from Gd 2 O 2 S:Tb is approximately monochromatic, and hence the variance in can E opt be ignored. A value of and E ab its associated variance were determined FIG. 4. The modulation transfer function MTF of: i the copper plate/ phosphor screen detector; ii the copper plate/phosphor screen due to the spread of high energy particles only; iii the copper plate/phosphor screen due to diffusion of optical quanta only; iv the lens; and, v the T.V. camera and lens assembly. The MTF s associated with the copper plate/ phosphor screen detector were measured using 6 MV x rays. The uncertainty in the MTF s ranged between 0.04 and from each AED in the following manner. Each AED was converted into probability density functions f E (E) by normalizing the area of each to one. The value of E and ab its variance are therefore given by the first moment about the origin i.e., 0 E ab Ef E (E)dE] and the second moment 2 about the mean i.e., Eab (E )2 E ab f E (E)dE], respectively. The ten values of E and ab Eab were subse- 2 quently averaged. The average value of 0.44 E ab MeV and its variance MeV 2 ) were used in Eq. 6, resulting in g and 2 g This surprisingly large value of g3 2 is due to the very large variance in E. ab The corresponding Poisson excess is g3 ( 2 g3 /g 3 ) The high value of g3 suggests that, when high-energy x rays are used, the energy absorption noise, which is related to the noise associated with the generation of optical quanta, is not Poisson noise. However, it is shown in the results section that this large excess value does not degrade the overall DQE of the T.V. camera-based system significantly. Note that Eq. 6 ignores variations in light output due to spatial variations in energy deposition and light transport in phosphor screens of finite thickness. 1 Because of the high energy of the incident x rays, x-ray interactions in the x-ray detector do not deposit all of their energy locally. Thus, the situation at megavoltage energies is much different than that encountered in diagnostic radiology Stage 4: Spread of optical quanta in the phosphor screen, T 4 ( ) The fourth stage is characterized by the spread of optical quanta in the phosphor screen T 4 ( ). It is not possible to measure T 4 ( ) directly. Rather, only the MTF of the copper plate/phosphor screen combination, T 2 ( ) T 4 ( ) including spread due to radiation and optical processes, and T 2 ( ) the MTF due to radiation processes can be measured

6 820 Bissonnette et al.: Quantum accounting and detective quantum efficiency 820 directly. The product T 2 ( ) T 4 ( ) has been measured previously, 11 using a method similar to that described in Sec. IV A 2. The value of T 4 ( ) is therefore obtained by dividing the experimentally measured MTF of the copper plate/ phosphor screen for the 6 MV x-ray beam from a linear accelerator Varian 2100c by T 2 ( ). 5. Stage 5: Escape of optical quanta from the phosphor screen, g 5 The fifth stage involves the probability (g 5 ) that an optical quanta generated inside the phosphor escapes through the exit surface of the phosphor. For our 360 mg/cm 2 phosphor, the value of g 5 has been calculated using the Kubelka Munk theory 24 to be 0.4 David Trauernicht, Eastman Kodak, private communication. Because of the difference between the value of g 5 given by the screen manufacturer i.e., g 5 0.4) and that reported from Monte Carlo simulations of optical light transport inside a phosphor screen of thickness similar to ours i.e., g 5 0.2), 22 the uncertainty in g 5 has been estimated to be Stage 6: Collection of light quanta by the lens, g 6 The sixth stage involves the lens collection efficiency (g 6 ) of the imaging system, which is the fraction of photons exiting the phosphor that reach the target of the T.V. camera. The lens collection efficiency, estimated by taking the ratio of the area of the lens of the T.V. camera to half surface area of the sphere where light exiting the phosphor is emitted, can be expressed as 25 r g 6 4 f 2 1 m 2, 7 where f is the F-number of the lens, m is the image demagnification factor, and is the transmittance of the lens. The lens of our T.V. camera Fujinon, model CF50L, NJ has a F-number of 0.7, and we have assumed a transmittance of 0.9. For our geometry, the value of m is The resulting value of g 6 is Stage 7: Spread of light quanta in the lens, T 7 ( ) The optical spreading process in the seventh stage is described by the MTF of the lens T 7 ( ). We can measure the combined MTF of the lens and T.V. camera assembly i.e., T 7 ( ) T 9 ( )]. However, it is essential to separate this measured MTF into the MTF s of the two individual components in order to calculate the quantum accounting diagram. In addition, this separation gives the opportunity to see how the QAD and the DQE change when we consider different combinations of lenses and T.V. cameras. Therefore, the MTF of the lens was obtained by dividing the measured T 7 ( ) T 9 ( ) by the MTF of the T.V. camera only T 9 ( ). T 9 ( ) was obtained from the data sheet of the XQ2182 tube of the T.V. camera Philips Components, Slatersville, RI. Note that the tube MTF reported by the manufacturer is, in fact, the product of the MTF of the tube and the MTF of the lens used in the measurements. However, these measurements are performed using a small lens aperture i.e., F number of 5.6, which reduces the loss in spatial resolution due to the lens. Therefore, we have assumed that the reported camera MTF is entirely due to the camera itself. The determination of the MTF of the lens and T.V. camera assembly T 7 ( ) T 9 ( ) involved the measurement of the LSF of the lens and T.V. camera assembly, 26,27 using standard image acquisition techniques. 11,13,21 Two slabs of black plastic were glued together to form a narrow 85 m wide, 5 cm long light beam. These slabs were placed at the same location normally occupied by the x-ray detector, at the center of the field of view of the T.V. camera. A total of 60 images of this narrow light beam were acquired, slit images and another 60 images were acquired with no light reaching the T.V. camera background images. Each set of images was summed to reduce noise. The difference between the summed slit images and the summed background images resulted in the LSF image, which was used to generate the LSF and which was Fourier-transformed to yield T 7 ( ) T 9 ( ). The two sets of images were acquired by accumulating signals on the target of the T.V. camera for 1 s, which is the exposure time required to acquire clinical images, in order to ensure that time-dependent phenomena had similar effects on both the MTF measurements and the clinical images. 11 These measurements were performed with the narrow light beam placed at a slight angle from the direction parallel to the T.V. camera scan lines in order to improve the sampling of the LSF 28 and to prevent asymmetries in the measured LSF. 11 The measurement of T 7 ( ) T 9 ( ) was confirmed by measuring the square-wave response of the lens and T.V. camera assembly, which was converted to T 7 ( ) T 9 ( ) using the theoretical relationship derived by Coltman. 29 In order to mimic the light emitted by the copper plate phosphor screen detector, all of these measurements were performed with ambient light filtered by a green filter Eastman Kodak, Wratten filter #74, Rochester, NY mounted on the lens of the T.V. camera. 8. Stage 8: Detection of optical quanta by the T.V. camera, g 8 The eighth stage is described by the detection efficiency (g 8 ) of the T.V. camera, which is the probability that an incident light quantum releases a photoelectron in the target of the T.V. camera. The value of g 8, obtained by weighing the spectral response of the XQ2182 tube of the T.V. camera by the spectrum of light emitted by Gd 2 O 2 S:Tb, 30 is The spectral response of the XQ2182 tube was obtained from the tube s data sheet. 9. Stage 9: Spread in the T.V. camera, T 9 ( ) The ninth stage involves the spreading process associated with the T.V. camera only T 9 ( ). As mentioned in Sec. IVA7, T 9 ( ) was obtained from the data sheet of the XQ2182 tube of the T.V. camera.

7 821 Bissonnette et al.: Quantum accounting and detective quantum efficiency 821 TABLE I. Summary of the gains and Poisson excesses for the eight stages of the T.V. camera-based portal imaging system. For gain stages, T i ( ) 1.0. Stage # Description Type of process Symbol Gain/efficiency (g i ) Poisson excess gi 1 Absorption of x-ray energy in phosphor Binomial g Radiative spread in phosphor Spread T 2 ( ) Generation of optical quanta in phosphor Amplification g Optical spread in phosphor Spread T 4 ( ) Escape probability of optical quanta Binomial g Collection by lens Binomial g MTF of lens Spread T 7 ( ) Detection in the T.V. camera Binomial g MTF of T.V. camera Spread T 9 ( ) Noise added by T.V. camera electronics S a ( )/ Stage 10: Additive noise, S a ( ) The last stage involved the addition of noise by the electronics of the T.V. camera, described by the term S a ( )/ in Eq. 4. The calculation of this additive term was performed using the NPS measured while no light was allowed to reach the lens of the T.V. camera. This NPS was measured using the two-dimensional technique described previously. 11 Briefly, a large number of independent images were acquired and subtracted in pairs to remove any structural variations in the images. The two-dimensional NPS was calculated by averaging the modulus of the Fourier transform of the central portion pixels of these subtracted images. Since the subtraction of statistically independent images doubles the variance, the NPS was divided by 2. 11,14,31 The slice through the two-dimensional NPS corresponding to the direction perpendicular to the camera scan lines was selected. The unitless term S a ( )/ needed for the QAD analysis was obtained by dividing the NPS by the average pixel value of the images obtained with a one monitor unit irradiation, the number of ADC units per electron generated in the camera, and the pixel area scaled to the plane of the x-ray detector. The value of S a ( )/ ranged between 2.2 and 13.6, depending on spatial frequency. The fluence-to-dose conversion factor, obtained by weighing fluence-to-dose conversion factors 32 with the 6 MV x-ray spectrum of Kubsad et al. 16 was cgy mm 2 /x ray at the isocenter of the linear accelerator, at a depth of 5 cm in water. Stage 10 describes the addition of noise with no quantum gain nor stochastic blur. This corresponds to g 10 1, g10 1, and T 10 ( ) 1. B. Calculation of the DQE 1. Experimental calculation The experimental DQE of the portal imaging system was first determined using Eq. 3 and previously measured MTF and NPS. 11 The DQE was calculated with the slices through the two-dimensional MTF and NPS which correspond to the direction perpendicular to the camera scan lines. 2. Theoretical calculation The spatial-frequency dependent DQE of the T.V. camera-based portal imaging system is obtained using Eq. 4 and the values of the gains, Poisson excesses, and relative additive NPS summarized in Tables I and II. Many terms in Eq. 4 cancel, and the resulting DQE can be simplified to yield g 1 T 2 2 DQE 1 g3 /g 3 1 S a / / g 3 g 5 g 6 g 8 T 2 4 T 2 7 T The product T 7 ( ) T 9 ( ) corresponds to the MTF measured for the lens and T.V. camera assembly. Thus, it is not actually necessary to determine the MTF of each separately for the DQE, as only the MTF of the lens and T.V. camera combination which was measured directly is required in Eq. 8. It is necessary, however, to determine T 7 ( ) and T 9 ( ) separately to obtain the QAD of the system, which may reveal secondary quantum sinks in these stages. V. RESULTS A. Absorbed energy distribution The AED for the x-ray detector i.e., 1 mm copper plate with a 360 mg/cm 2 thick gadolinium oxysulfide layer and the 6 MV spectrum of Kubsad et al. 16 is shown in Fig. 3. From this AED, the probability that incident x rays deposit energy in the phosphor layer of the x-ray detector, g 1, the

8 822 Bissonnette et al.: Quantum accounting and detective quantum efficiency 822 TABLE II. Values of the relative additive NPS, S a ( )/, at five different spatial frequencies. Spatial frequency S a ( )/ gain associated from the conversion of x-ray quanta to optical quanta, g 3, and the Poisson excess associated with the latter gain, g3, were determined as described in Secs. IVA1andIVA3. The AED shown in Fig. 3 shows discontinuities near 0.08 and 0.25 MeV. The discontinuity at 0.08 MeV is due to three sources: i The x-ray spectrum of Kubsad et al., 16 which suggests that x-ray photons of energies between 0 and 0.25 MeV are equally probable; ii the filtering of the incident beam by the copper plate there is an abrupt increase in the absorption of x rays by the 1 mm copper plate as the energy of the incident x rays decreases from 0.1 to 0.05 MeV; and, iii the K-edge of gadolinium. Thus, the incident spectrum contains photons ranging in energy between 0 and 0.25 MeV, the lower energy photons of this spectrum are preferentially absorbed in the copper plate, and these photons reaching the screen undergo primarily photoelectric interactions in the screen, resulting in the K-edge of gadolinium influencing the absorption process. The discontinuity at 0.25 MeV is caused by the low energy resolution ( E 0.25 MeV of the x-ray spectrum used in our Monte Carlo simulations, 16 which, in turn, causes an over-representation of x-ray energies just above 0.25 MeV and an underrepresentation of energies just below 0.25 MeV. Since most incident x rays with energy below 0.35 MeV deposit energy in the phosphor layer of the x-ray detector through photoelectric interactions, discontinuities in the incident x-ray spectrum result in the underestimation of the height of photopeaks below 0.25 MeV and an overestimation of the height of photopeaks above 0.25 MeV. This causes a spurious peak in the AED at 0.25 MeV, caused by the input spectrum. Note that the discontinuity at 0.25 MeV does not dominate the shape of the AED shown in Fig. 3, and therefore has a negligible effect on the QAD parameters extracted from the AED. B. Measurements Figure 4 shows the MTF s used for the QAD analysis of the video-based portal imaging system. The error bars represent one standard deviation from the associated average MTF. Figure 4 shows that the loss in spatial resolution due to the transport of high-energy particles T 2 ( ) is less than that caused by optical diffusion in the phosphor screen T 4 ( ). Note that the T 2 ( )shown in Fig. 4 is probably an underestimate of the MTF due to the spread of high-energy radiation since the phosphor screen and the EM-1 singleemulsion film used to record the signals were separated by a thin, low-density, opaque plastic film. The high energy particles can spread slightly while passing through the opaque plastic film. Figure 4 also shows that the MTF of the lens FIG. 5. Spatial-frequency dependent quantum accounting diagram QAD of the T.V. camera-based portal imaging system. The shaded area spans the stages involved with the x-ray detector of the imaging system. T 7 ( ) is much worse than the MTF of the camera tube T 9 ( ). Note that the value of T 7 ( ), which was determined using the method described in Sec. IV A 7, agrees well with the limited MTF data supplied to us by the lens manufacturer. Since the MTF of the T.V. camera tube supplied by the tube manufacturer is much higher than the lens MTF, Fig. 4 suggests that there is more potential for spatial resolution improvements with the lens than with the T.V. camera tube. The frequency-dependent QAD of the portal imaging system is shown in Fig. 5, and the values associated with the QAD lines are shown in Table III. In Fig. 5, the solid line represents the conventional zero-frequency QAD while the broken lines represent spatial-frequency-dependent values. It can be seen from Fig. 5 that, for many nonzero spatial frequencies, the number of optical quanta contributing to the image i.e., at the last stage is lower than the number of x-rays which deposit energy in the phosphor layer of the copper plate/phosphor screen detector. Therefore, the portal imaging system is not x-ray quantum limited over many frequencies of interest. In addition, in the final image-forming stage, the zero-frequency QAD is only slightly greater than the number of quanta at the x-ray detection stage, while the QAD at higher spatial frequencies is significantly less. The divergence between the QAD lines due to spreading processes, especially by T 4 ( ) and T 7 ( ), is shown clearly. Also, Fig. 5 shows that the most important cause of signal loss is the poor lens collection efficiency of the imaging system, and that the MTF of the lens significantly degrades image quality at high spatial frequencies. Note that the QAD is approximate since it does not account for noise added by electronic components of the imaging system, nor does it account for noise due to the Poisson excess in the gains. Figure 6 shows the DQE obtained with the QAD approach using Eq. 8 dotted line as well as the DQE calculated using Eq. 3 and previously published data solid line. In contrast to the QAD values, this result does account for

9 823 Bissonnette et al.: Quantum accounting and detective quantum efficiency 823 TABLE III. Values of the spatial-frequency dependent quantum accounting diagram QAD of the T.V. camera-based portal imaging system. The QAD is shown in Fig. 5. P i ( ) Stage # Symbol g T 2 ( ) g T 4 ( ) g g T 7 ( ) g T 9 ( ) FIG. 6. The detective quantum efficiency DQE of the portal imaging system. The dotted line was obtained from the QAD analysis of the portal imaging system, the dashed line was obtained from the QAD analysis after the effect of additive and gain noise. Good agreement, over a range of 2.5 orders of magnitude, exists between the calculated and the measured DQE s. The error bars in the experimental DQE s were obtained by propagating the uncertainties in the MTF and NPS of Bissonnette et al. 11 using the method described by Bevington. 33 Similarly, the error bars in the theoretical DQE were obtained by propagating the uncertainties in the gains and MTF s reported in this paper. The dashed line in Fig. 6 shows the theoretical DQE obtained if the additive noise term of Eq. 8, S a ( )/, is set to 0. Comparison of the dashed line in Fig. 6 with the solid line shows that a significant improvement in the DQE could be obtained if the additive noise could be eliminated. C. Parameter sensitivity analysis the additive noise term was set to 0, and the solid line was obtained from previously measured data Ref. 11. The error bars represent the standard deviation in the measured DQE. The expression derived for the DQE of the T.V. camerabased portal imaging system Eq. 8 depends on ten parameters, some of which depend on assumptions. In this section, we identify which parameters affect the DQE significantly, and what uncertainties may be acceptable in these parameters without affecting the conclusions of the analysis. Changing the value of the gain factors listed in Table I results in a greater improvement of the DQE obtained from Eq. 8 at high spatial frequencies than at low spatial frequencies. For example, doubling the absorption of x-ray energy in the phosphor screen (g 1 ) would increase the DQE by a factor of 2.4 at low spatial frequencies and 3.7 at high spatial frequencies. Doubling g 3 would increase the DQE by a factor ranging between 1.8 at low spatial frequencies and 3.7 at high spatial frequencies, and doubling any of the other gains would increase the DQE by a factor ranging between 1.4 at low spatial frequencies and 3.7 at high spatial frequencies. The improvement in the DQE can be higher than the increase in the gain since the actual number of quanta in the last image-forming stage increases and, furthermore, the relative additive NPS i.e., S a ( )/ )] is reduced. Were the relative additive NPS negligible, the maximal increase in DQE would not exceed the increase in the gain factor. Note that, as shown in Fig. 6, reducing S a ( )/ to zero while maintaining all gain factors as listed in Table I would increase the DQE by a factor ranging between 1.5 at low spatial frequencies and 14.2 at high spatial frequencies. Also, note that g 1, g 5, g 6, and g 8 cannot be greater than one. The other source of non-poisson noise is at the stage where optical quanta are generated, and is accounted for by the Poisson excess g3. Surprisingly, if this noise was Poisson noise i.e., 2 g3 g 3, and g3 0), the DQE would increase only by a factor of 1.46 at 0 cycle/mm, and would not change considerably i.e., less than 5% for spatial frequencies higher than 0.5 cycle/mm. The modest influence of g3 on the system DQE is due to the secondary quantum sinks, which dominate at high spatial frequencies see Fig. 6. Consequently, the value of the right-hand-side term in the denominator of Eq. 8 is much larger than the other terms in the denominator, thereby reducing the relative importance of g3 on the system DQE. The MTF s associated with spreading stages also affect the DQE. Table IV shows the improvements to the DQE if the frequency at which each MTF is 0.5, f 0.5, is increased by 10%. Clearly, such a modest improvement in the MTF of the lens T 7 ( ) or the MTF of the lens and camera combination T 7 ( ) T 9 ( ) is sufficient to improve the DQE at high spatial frequencies. In contrast, similar changes in the other MTF s only have a moderate influence on the DQE at high spatial frequencies. These results show that the MTF of the lens is a major determinant of the spatial resolution of the T.V. camera-based imaging system. To summarize, the parameter sensitivity analysis shows that, if the individual gains are increased by identical factors, the DQE is most affected by changes in the x-ray detection probability, g 1.The DQE is less sensitive to changes in other gain stages at low spatial frequencies, and modestly sensitive to changes in g3. As far as increments of the MTF s are concerned, the DQE at high spatial frequencies is considerably affected by changes in the lens MTF T 7 ( ), and only moderately affected by changes in the other MTF s

10 824 Bissonnette et al.: Quantum accounting and detective quantum efficiency 824 TABLE IV. Relative improvement of the DQE at four different spatial frequencies, resulting from an increase by 10% in the spatial frequency at which individual MTF s are 0.5. The greatest improvement is obtained by increasing the MTF of the lens and T.V. camera combination, T 7 ( ) T 9 ( ). 0 cycles/mm 0.25 cycles/mm 0.5 cycles/mm 0.74 cycles/mm T 2 ( ) T 4 ( ) T 7 ( ) T 9 ( ) T 7 ( ) T 9 ( ) T 2 ( ), T 4 ( ), and T 9 ( )]. Also, the DQE can be increased considerably if additive noise can be eliminated. VI. DISCUSSION This paper presents the first experimental demonstration of the QAD theory. Only Monte Carlo simulations have been used previously. 3,9,12 The DQE calculated using the QAD appears to be slightly greater than the measured DQE; however, they agree within statistical uncertainty, giving confidence that the QAD analysis shown in this paper can be used as a tool to analyze the performance of T.V. camera-based portal imaging systems and to suggest what modifications should improve image quality. Note that our analysis applies only to the central region of the video-based portal imaging system since the system is not shift-invariant over the entire field of view. Fortunately, linearity and shift-invariance can be reasonably assumed for the central quadrant of images obtained with our system. The expression derived for the DQE presented in this paper, Eq. 8, complements a previous zero-spatial frequency DQE derived in Munro et al. 10 for video-based portal imaging systems. In Ref. 10, the zero-spatial frequency DQE was derived assuming that all processes are either Poissondistributed gains or binary selections. In Eq. 1 of Ref. 10, P 1 corresponds to our g 1 ; P 2 to the product of our g 3 and g 5 ; P 3 to our g 6 ; and, P 4 to our g 8. Equation 8 is consistent with Ref. 10, but represents a more comprehensive derivation which accounts for non-poisson noise and evaluates the DQE for all spatial frequencies. Equation 8 demonstrates that maximizing spatial resolution i.e., the MTF s and minimizing non-poisson noise i.e., g3 and S a ( )/ ]is necessary to maximize the DQE. However, the implications from the zero-spatial frequency DQE derived by Munro et al. 10 still hold: to maximize the DQE, the product of the gains and squared MTF s i.e., P i ( )] should be much greater than unity. The main limitation to the QAD analysis is the uncertainty in some of the gain factors involved. The largest uncertainty is in the value of the probability that an optical photon generated inside the phosphor screen exits through the exit surface of the screen (g 5 ): the value of g 5 used in this analysis is twice as large as that determined by Radcliffe et al. 22 for a screen thickness similar to that of our x-ray detector. The QAD plot Fig. 5 indicates that the T.V. camerabased portal imaging system is not quantum-noise limited. This result confirms the conclusions of an earlier study. 11 Note that these results agree with those obtained by Mah et al., 31 where they suggest that x-ray quantum noise can only be observed when the lens collection efficiency is dramatically increased. To remove these secondary quantum sinks, P i ( )increases of at least 8.7 and 195 are required to ensure that the primary quantum sink dominates for 0.5 and 0.74 cycles/mm spatial frequencies, respectively. This can be achieved by increasing the gain factors or MTF s in the optical chain of the imaging system. However, changes in the design of the T.V. camera-based system would have difficulty eliminating all of the secondary quantum sinks. The QAD analysis shows that the lens collection efficiency, g 6, is the major cause of signal loss. The value of g 6 can be increased by increasing the area of the light sensor. Replacing the current T.V. camera useful target area of 2.06 cm 2 ) by a large-area charge-coupled device CCD camera useful target area of cm 2 has been reported Dalsa Megasensor CA-D9-5120, Waterloo, Ontario would decrease the demagnification factor involved in the calculation of g 6, resulting in an increase of P i ( ) by a factor of about 18, assuming no changes in other gains or MTF s. However, a large-area CCD may not be practical because of the large size of the lens required to cover such a large area light sensor. Furthermore, this increase, in combination with other changes in the current system design, is unlikely to remove the secondary quantum sinks for the highest spatial frequencies 0.7 cycle/mm where an improvement factor of 195 is required. While eliminating additive noise and improving the MTF of the lens and T.V. camera combination is likely to have a significant and beneficial effect on image quality, it is unlikely for video-based portal imaging systems to eliminate all secondary quantum sinks for spatial frequencies above 0.5 cycle/mm. The parameter sensitivity analysis performed on the derived expression for the DQE of the portal imaging system shows that the DQE can be affected, in order of decreasing efficiency, by changes in: i The detection efficiency, g 1 ; ii S a ( )/ ; iii the MTF of the lens and T.V. camera assembly, T 7 ( ) T 9 ( ); iv the optical gain, g 3 ; v either of g 5, g 6,org 8 ; and, vi the MTF of the x-ray detector, T 2 ( ) T 4 ( ).Conversely, the noise associated with the optical gain affects the DQE of the current system only minimally. Therefore, improving the x-ray detector detection efficiency is the most efficient way to improve the current system, even though the lens collection efficiency is the most important cause of signal loss. Nonetheless, research groups are investigating the use of solid-state technology to eliminate the losses due to the poor lens collection efficiency. Already, the portal imaging system based on an amorphous silicon detector shows promise of significant improvements in QAD and DQE. 34